Optimal. Leaf size=40 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (2-x^2\right )}{2 \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.0268767, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1114, 724, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (2-x^2\right )}{2 \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{3-3 x^2+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3-3 x+x^2}} \, dx,x,x^2\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{3 \left (2-x^2\right )}{\sqrt{3-3 x^2+x^4}}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (2-x^2\right )}{2 \sqrt{3-3 x^2+x^4}}\right )}{2 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0091794, size = 40, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\frac{ \left ( -3\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}-3\,{x}^{2}+3}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6456, size = 27, normalized size = 0.68 \begin{align*} -\frac{1}{6} \, \sqrt{3} \operatorname{arsinh}\left (-\sqrt{3} + \frac{2 \, \sqrt{3}}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31135, size = 131, normalized size = 3.28 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{3 \, x^{2} + 2 \, \sqrt{3}{\left (x^{2} - 2\right )} + 2 \, \sqrt{x^{4} - 3 \, x^{2} + 3}{\left (\sqrt{3} + 2\right )} - 6}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{x^{4} - 3 x^{2} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{4} - 3 \, x^{2} + 3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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